field of quotients

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field of fractions

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field of quotients (plural fields of quotients)

1. (algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: ${\displaystyle (a,b)+(a',b')=(ab'+a'b,bb')}$, the multiplicative operator is defined coordinate-wise, the zero is ${\displaystyle (0,1)}$, the unity is ${\displaystyle (1,1)}$, the additive inverse of ${\displaystyle (a,b)}$ is ${\displaystyle (-a,b)}$, equivalence is defined like so: ${\displaystyle (a,b)\equiv (a',b')}$ if and only if ${\displaystyle ab'=a'b}$, and multiplicative inverse of a non-zero–equivalent element ${\displaystyle (a,b)}$ is ${\displaystyle (b,a)}$.

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